Comparison and Uniqueness for the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$, $f$ and the affine data $b_{\mu}$ satisfy the hypotheses of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure, and let $(\mathrm{LQ})$ be the linear-quadratic Hamilton--Jacobi equation with discount rate $\rho$, drift $b$ and running cost $f$. Metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

1. (Comparison)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, let $u$ be upper semicontinuous and a plan-jet viscosity subsolution of $(\mathrm{LQ})$, let $v$ be lower semicontinuous and a plan-jet viscosity supersolution of $(\mathrm{LQ})$. Then $u(\mu)\le v(\mu)$ for every $\mu\in\Sigma^{2}_{d}$.

2. (Uniqueness)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, continuous plan-jet viscosity solutions of $(\mathrm{LQ})$. Then $u=v$.